Question: $y'=2x+3y-1$ Is $y=-\dfrac23x+\dfrac19$ a solution to the above equation? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Answer: In order to find whether $y=-\dfrac23x+\dfrac19$ is a solution, we need to substitute it into the equation and see if we get equivalent expressions on each side of the equal sign. In addition to substituting for $y$, we need to find the corresponding $y'$ expression to substitute into the equation: $\begin{aligned} y'&=\dfrac{d}{dx}\left[-\dfrac23 x+\dfrac19\right] \\\\ &=-\dfrac23 \end{aligned}$ Now we substitute ${y=-\dfrac23x+\dfrac19}$ and ${y'=-\dfrac23}$ into the equation: $\begin{aligned} {y'}&=2x+3{y}-1 \\\\ {-\dfrac23}&\stackrel{?}{=}2x+3\left({-\dfrac23x+\dfrac19}\right)-1 \\\\ -\dfrac23&\stackrel{?}{=}2x-2x+\dfrac13-1 \\\\ -\dfrac23&\stackrel{\checkmark}{=}-\dfrac23 \end{aligned}$ We obtained the same expression on each side. In conclusion, yes, $y=-\dfrac23x+\dfrac19$ is a solution to the differential equation.